My grandfather used to smoke soprasatta and each year we would visit and bring home about a dozen sticks. My dad had a huge ceramic crock in the basement that was filled with olive oil and he stored the sausage in that.

Put it in the fridge in a paper bag. There it will dry further, and will certainly not get mushy. The opposite will happen. If you like it somewhat softer, wrap it in foil before placing it in the fridge. I'm not partial to the oil technique, sorry.

You didn't say what type of sausage, but if it's white on the outside that's a beneficial penicillium mold that was deliberately added and will protect against nasty green or black molds. If it has a plain dried casing and happens to develop some white mold, don't panic. Let it go or rub the sausage down with a little vinegar. Black or green mold is a bad sign. Time for major surgery or disposal if it gets really bad.

As Cheese Boy says, just put it in the fridge, covered with foil if you like. I never heard of storing cured meats in oil. Sounds like an invitation to botulina if the meat was cured without nitrates.